Of course Irenaeus would have held this view and I don't disagree with Irenaeus' writings. There was only ONE Christian church at that time and it was under attacked by all sorts of heretical doctrine. Irenaeus focus was on keeping the Church pure and his writings reflect this view.
Shoot ahead 900 years later when the Orthodox split. Which traditions are you following? The Orthodox says that the Church never had a strong Pope. The Roman Catholics say they did. This was such an area of contention it has kept both groups apart for another 1,000 years. Well, if you're following traditions of the Church doesn't this seem like a silly argument? One would have thought the matter resolved. Whose tradition is it anyway?
So obviously, I would say that tradition in its entirety rests in the Catholic church, and that the Orthodox Churches preserve almost all of it intact (but not quite all of it, because of the non-communion with the See of Peter).
Sometime ask the Orthodox the question Jesus asked Peter, "Who do men say that I am? ... Who do you say that I am?" in reference, not to Jesus, but to the Pope.
There are almost as many answers as there are Orthodox.
I think the problem is that our understanding of what the Roman primacy means has diverged, and it was diverging for quite some time before 1054. So, some of the Orthodox might say that would be happy to recognize a Roman primacy that operated they way they think the Roman primacy operated before, say AD 800. Problem: even if we could understand accurately how the east viewed the Roman primacy before AD 800 and reproduce it today, that's not necessarily the way the West viewed it before AD 800, to say nothing of the way the West views it today.
Overlapping (not identical) tradition, but different ways of understanding it, especially in the area of ecclesiology and church government.